Exam 18: Simple Linear Regression and Correlation

A statistician investigating the relationship between the amount of precipitation (in inches) and the number of car accidents gathered data for 10 randomly selected days. The results are presented below. Day Precipitation Number of accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 8 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 Estimate with 95% confidence the mean daily number of accidents when the daily precipitation is 0.25 inches.

A financier whose specialty is investing in movie productions has observed that, in general, movies with 'big-name' stars seem to generate more revenue than those movies whose stars are less well known. To examine his belief, he records the gross revenue and the payment (in $ million) given to the two highest-paid performers in the movie for 10 recently released movies. Movie Cost of two highest- paid performers (\ ) Gross revenue (\ ) 1 5.3 48 2 7.2 65 3 1.3 18 4 1.8 20 5 3.5 31 6 2.6 26 7 8.0 73 8 2.4 23 9 4.5 39 10 6.7 58 Draw a scatter diagram of the data to determine whether a linear model appears to be appropriate.

The value of the sum of squares for regression, SSR, can never be smaller than 1.

At a recent music concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the beginning of the year. The following data were collected. Age 62 57 40 49 67 54 43 65 54 41 Number of concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 SUMMARY OUTPUT DESCRIPTIVE STATISTICS Reqression Statiatics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 Age Concerts Mean 53 Mean 3.65 Standard Error 2.1849 Standard Error 0.3424 Standard Deviation 9.7711 Standard Deviation 1.5313 Sample Variance 95.4737 Sample Variance 2.3447 Count 20 Count 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ ANOVA df SS MS F Significance F Regression 1 28.65711 28.65711 32.45653 2.1082-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 a. Predict with 95% confidence the number of concerts attended by a 45-year-old individual. b. Predict with 95% confidence the average number of concerts attended by all 45-year-old individuals.

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Conduct a test of the population slope to determine at the 5% significance level whether a linear relationship exists between TV game show contestants' years of education and their winnings.

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct? \begin{array}{|l|l|}\hline A&\text {There is a strong, positive linear relationship between \mathrm{x} and \( \mathrm{y} \), where there may or }\\&\text { may not be any causal relationship between \( x \) and \( y \).}\\\hline B&\text { There is a strong, negative linear relationship between \( \mathrm{x} \) and \( \mathrm{y} \), where there}\\&\text { must be a causal relationship between \( x \) and \( y \). }\\\hline C&\text {There is a weak, negative linear relationship between \( \mathrm{x} \) and \( \mathrm{y} \), where there may or }\\&\text {may not be any causal relationship between \( x \) and \( y \). }\\\hline D&\text {There is a strong negative linear relationship between \( \mathrm{x} \) and \( \mathrm{y} \), where there may or }\\&\text { may not be any causal relationship between \( x \) and \( y \).}\\\hline \end{array}

In simple linear regression, the coefficient of correlation r and the least squares estimate $b _ { 1 }$ of the population slope $\beta _ { 1 }$ : A. Must be the same size and sign. B. May have both have a different size and different sign. C. May be the same size but have different sign. D. May be different sizes but will have the same sign.

Which of the following best describes the value of the slope, if the coefficient of determination is 0.95? A. Slope must be 1 . B. Slope must be negative. C. Slope must be 0.95 . D. Slope must be positive.

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Use the predicted and actual values of y to calculate the residuals.

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. Book Number of pages Selling price (\ ) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 Estimate with 90% confidence the mean selling price of all books with 900 pages.

Plot the residuals against the predicted values of y. Does the variance appear to be constant?

An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief, she gathers data about the last eight winners of her favourite game show. She records their winnings in dollars and their years of education. The results are as follows. Contestant Years of education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 6 16 300 7 13 650 8 14 400 Compute the standardised residuals.

The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyse the claim, a university student visits the bookstore and records the number of pages and the selling price of 12 randomly selected books. These data are listed below. Book Number of pages Selling price (\ ) 1 844 55 2 727 50 3 360 35 4 915 60 5 295 30 6 706 50 7 410 40 8 905 53 9 1058 65 10 865 54 11 677 42 12 912 58 Can we infer at the 5% significance level that the editor is correct?

In a regression problem the following pairs (x, y) are given: (3,-2), (3,-1), (3,0), (3,1) and (3,2). This indicates that the coefficient of correlation is -1.

The value of the sum of squares for regression, SSR, can never be larger than the value of sum of squares for error, SSE.

Statisticians have shown that the sample y-intercept $b _ { 0 }$ and sample slope coefficient $b _ { 1 }$ are unbiased estimators of the population regression parameters $\beta _ { 0 }$ and $\beta _ { 1 }$ .

In simple linear regression, the divisor of the standard error of estimate, $S _ { \varepsilon }$ , is n - 1.

The symbol for the sample coefficient of correlation is: A r B \rho C D

If the coefficient of correlation is 0.80, the percentage of the variation in y that is explained by the variation in x is: A. 80\% B. 0.64\% C. -80\% D. 64\%

The quality of oil is measured in API gravity degrees - the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field, who believes that there is a relationship between quality and price per barrel. Oil degrees API Price per barrel (in \ ) 27.0 12.02 28.5 12.04 30.8 12.32 31.3 12.27 31.9 12.49 34.5 12.70 34.0 12.80 34.7 13.00 37.0 13.00 41.0 13.17 41.0 13.19 38.8 13.22 39.3 13.27 A partial Minitab output follows. Descriptive Statistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Frice 13 12.730 0.457 0.127 Covariances Degrees Price Degrees 21.281667 Price 2.026750 0.208833 Regression Analysis Fredictor Coef StDev Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S = 0.1314 R-Sq = 92.46% R-Sq(adj) = 91.7% Analysis of Variance Source DF SS MS F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 a. Determine the standard error of estimate and describe what this statistic tells you. b. Determine the coefficient of determination and discuss what its value tells you about the two variables. c. Calculate the Pearson correlation coefficient. What sign does it have? Why?